In this section we prove Theorem REF . As mentioned in the introduction, the positive part of our result will follow from Theorem REF  and ideas from Marks and Unger's Borel circle squaring proof [1]}. In the negative direction, the novelty is the identification of \((\mathbb {Z}\slash 2\mathbb {Z}) *(\mathbb {Z}\slash 2\mathbb {Z})\)  as essentially the only group that does not satisfy this strengthening of the Gardner conjecture.
